Complex Numbers from A to... Z
It is impossible to imagine modern mathematics without complex numbers. Complex Numbers from A to . . . Z introduces the reader to this fascinating subject that, from the time of L. Euler, has become one of the most utilized ideas in mathematics. The expo
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ex Numbers from A to. . . Z
Birkh¨auser Boston • Basel • Berlin
Titu Andreescu University of Texas at Dallas School of Natural Sciences and Mathematics Richardson, TX 75083 U.S.A.
Dorin Andrica “Babes¸-Bolyai” University Faculty of Mathematics 3400 Cluj-Napoca Romania
Cover design by Mary Burgess. Mathematics Subject Classification (2000): 00A05, 00A07, 30-99, 30A99, 97U40
Library of Congress Cataloging-in-Publication Data Andreescu, Titu, 1956Complex numbers from A to–Z / Titu Andreescu, Dorin Andrica. p. cm. “Partly based on a Romanian version . . . preserving the title. . . and about 35% of the text”–Pref. Includes bibliographical references and index. ISBN 0-8176-4326-5 (acid-free paper) 1. Numbers, Complex. I. Andrica, D. (Dorin) II. Andrica, D. (Dorin) Numere complexe QA255.A558 2004 512.7’88–dc22 2004051907
ISBN-10 0-8176-4326-5 ISBN-13 978-0-8176-4326-3
eISBN 0-8176-4449-0
Printed on acid-free paper.
c 2006 Birkh¨auser Boston
Complex Numbers from A to. . . Z is a greatly expanded and substantially enhanced version of the Romanian edition, Numere complexe de la A la. . . Z, S.C. Editura Millenium S.R. L., Alba Iulia, Romania, 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 www.birkhauser.com
(TXQ/MP)
The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard
About the Authors Titu Andreescu received his BA, MS, and PhD from the West University of Timisoara, Romania. The topic of his doctoral dissertation was “Research on Diophantine Analysis and Applications.” Professor Andreescu currently teaches at the University of Texas at Dallas. Titu is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002), Director of the Mathematical Olympiad Summer Program (1995–2002) and leader of the USA IMO Team (1995–2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world’s most prestigious mathematics competition. Titu received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a “Certificate of Appreciation” from the president of the MAA in 1995 for his outstanding servic
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